The Stacks project

Lemma 43.14.2. Let $X$ be a nonsingular variety. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. The $\mathcal{O}_ X$-module $\text{Tor}_ p^{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is coherent, has stalk at $x$ equal to $\text{Tor}_ p^{\mathcal{O}_{X, x}}(\mathcal{F}_ x, \mathcal{G}_ x)$, is supported on $\text{Supp}(\mathcal{F}) \cap \text{Supp}(\mathcal{G})$, and is nonzero only for $p \in \{ 0, \ldots , \dim (X)\} $.

Proof. The result on stalks was discussed above and it implies the support condition. The $\text{Tor}$'s are coherent by Lemma 43.14.1. The vanishing of negative $\text{Tor}$'s is immediate from the construction. The vanishing of $\text{Tor}_ p$ for $p > \dim (X)$ can be seen as follows: the local rings $\mathcal{O}_{X, x}$ are regular (as $X$ is nonsingular) of dimension $\leq \dim (X)$ (Algebra, Lemma 10.116.1), hence $\mathcal{O}_{X, x}$ has finite global dimension $\leq \dim (X)$ (Algebra, Lemma 10.110.8) which implies that $\text{Tor}$-groups of modules vanish beyond the dimension (More on Algebra, Lemma 15.66.19). $\square$


Comments (2)

Comment #2779 by Ko Aoki on

Typo in the proof: "he local rings" should be replaced by "the local rings".


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AZT. Beware of the difference between the letter 'O' and the digit '0'.